$ F = \left[\begin{array}{rrr}4 & 2 & 0 \\ -2 & 5 & 0\end{array}\right]$ $ w = \left[\begin{array}{r}4 \\ -1 \\ 1\end{array}\right]$ What is $ F w$ ?
Solution: Because $ F$ has dimensions $(2\times3)$ and $ w$ has dimensions $(3\times1)$ , the answer matrix will have dimensions $(2\times1)$ $ F w = \left[\begin{array}{rrr}{4} & {2} & {0} \\ {-2} & {5} & {0}\end{array}\right] \left[\begin{array}{r}{4} \\ {-1} \\ {1}\end{array}\right] = \left[\begin{array}{r}? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ w$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ w$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ w$ , and so on. Add the products together. $ \left[\begin{array}{r}{4}\cdot{4}+{2}\cdot{-1}+{0}\cdot{1} \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ w$ and add the products together. $ \left[\begin{array}{r}{4}\cdot{4}+{2}\cdot{-1}+{0}\cdot{1} \\ {-2}\cdot{4}+{5}\cdot{-1}+{0}\cdot{1}\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{4}\cdot{4}+{2}\cdot{-1}+{0}\cdot{1} \\ {-2}\cdot{4}+{5}\cdot{-1}+{0}\cdot{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}14 \\ -13\end{array}\right] $